TRANSPORT UNDER ADVECTIVE TRAPPING

Juan J. Hidalgo1, Insa Neuweiler2, and Marco Dentz1

1IDAEA-CSIC, Barcelona, Spain

2Leibniz Universität Hannover, Hannover, Germany

c© AUTHORS. ALL RIGHTS RESERVED

INTRODUCTION

Motivation:I Advective trapping occurs when solute enters low velocity zones in

heterogeneous porous media.I Classical approaches combine slow advection and diffusion into a dispersion

coefficient or a single memory function.ObjectiveI We investigate advective trapping in homogeneous media with low

permeability circular inclusions.I We build an upscaled model in the continuous time random walk framework.

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FLOW PROPERTIES

Mean velocity in the matrix isproportional to the area occupied bythe inclusions χ.Velocity in the inclusions is notconstant. The mean velocity in theinclusions vi is log-normallydistributed and proportional to χ.

FIGURE 1: Streamlines in a mediumwith randomly placed inclusions.

0

150

300

0.01 0.02 0.03 0.04 0.05 0.06

p(v)

v

vi

vi

10−1

100

101

102

103

0.01 0.02 0.03 0.04 0.05 0.06 0.07

p(vi)

vi

r0r0/2r0/3r0/4

0

0.1

0.2

0.3

0.4

0.5

χ

FIGURE 2: Velocity distribution inside theinclusions (top) and mean velocitydistribution (bottom).c© AUTHORS. ALL RIGHTS RESERVED

TRANSPORT PROPERTIES

The breakthrough curves reflect thetrapping of particles in the lowpermeability inclusions.The trapping rate follows a Poissondistribution.

FIGURE 3: Transport through a mediumwith randomly placed inclusions.

10−5

10−4

10−3

10−2

10−1

100

0 10 20 30 40 50

f(t)

t

xc

ℓc5ℓc10ℓc

(a)

10−5

10−4

10−3

10−2

10−1

200 250 300 350 400 450 500 550 600

f(t)

t

xc

300ℓc387ℓc

(c)

10−4

10−3

10−2

10−1

100

0 5 10 15 20

p(ntr)

ntr

xc

10ℓc50ℓc200ℓc387ℓc

FIGURE 4: Breakthrough curves at increasingdistance from the inlet and advection -dispersion equation solution fit (top). Thenumber of trapping events follows a Poissondistribution (bottom).c© AUTHORS. ALL RIGHTS RESERVED

UPSCALED CONTINUOUS RANDOM TIME WALK MODEL I

We consider advective-dispersive particle transitions in the mobile matrix

dx(s) = vmatrixds +√

2Dmdsξ(s),

with s the mobile time spend outside the inclusions, Dm is diffusion (fromEames & Bush, 1999) and ξ(s) a Gaussian white noise.During the mobile time s particles encounter ns inclusions. The clock time t(s)after the mobile time s has passed is given by

t(s) = s +ns

∑i=1

τi

where ns is Poisson distributed and the trapping times τi depend on thedistance (random uniform) and the velocity at the visited inclusion (randomlog-normal)

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UPSCALED TRANSPORT MODEL RESULTS

10−5

10−4

10−3

10−2

10−1

100

0 10 20 30 40 50

f(t)

t

xc

ℓc5ℓc10ℓc

(a)

10−5

10−4

10−3

10−2

10−1

200 250 300 350 400 450 500 550 600

f(t)

t

xc

300ℓc387ℓc

(c)

FIGURE 5: Breakthrough curves at increasing distance from the inlet (dots) and upscaledCTRW model results 8solid line).

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SUMMARY & CONCLUSIONS

Purely advective transport was here considered as a limiting case foradvective-diffusive transport.The shape of breakthrough the curves cannot be predicted with amacrodispersion coefficient.We developed a CTRW model developed parameterized by measurablemedium properties: the trapping rate (Poisson distributed), the velocity inthe matrix (a function of χ) and the mean velocity distribution inside theinclusions (log-normal).

Acknowledgements

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